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\title{\phantom{.} \vskip-3cm
\huge The Sierpinski Triangle, The Sierpinski Curve 
\footnote{\large This file is from the 3D-XploreMath project. 
\hfil\break Please see http://www.math.uci.edu/$\sim$palais/  or http://3d-xplormath.org/}}
\author{}
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\maketitle
\vskip-2cm
\LARGE
The Sierpinski Triangle is a well known example of a ``large'' compact set without
interior points. It is defined by the following construction:

Start with an equilateral triangle and subdivide it into four congruent equilateral triangles.
Remove the middle one. Subdivide the remaining triangles again and remove in each
the middle one. Repeat this procedure. In each step the area is reduced by a factor $3/4$.
\vskip4mm \hrule \vskip6pt

\centerline{ In fact, the Sierpinski Triangle is the image of a continuous curve. }

\vskip4pt\hrule
As in the other fractal curves in 3DXM we have to define an iteratively defined and uniformly
convergent sequence of polygonal curves. As in the case of the Hilbert square filling curve
there is an easier construction by non-injective curves which, however, can be modified to give 
better looking injective approximations. In the following illustration we have chosen the 3DXM
parameter $bb$ -- which gives the Sierpinski Curve for $bb=0.5$ -- to be $bb=0.49$, because
in this case also the easier construction gives injective approximations (of course of a slightly
different limit curve).
\vskip15pt

\hbox{\hskip-20pt
\vbox{\hsize=0.33\hsize \includegraphics[width=2.2 in]{Sierpinski0.png}}
\vbox{\hsize=0.33\hsize \includegraphics[width=2.2 in]{Sierpinski1.png}}
\vbox{\hsize=0.33\hsize \includegraphics[width=2.2 in]{Sierpinski2.png}}
}

The starting polygonal curve has the vertices and the edge midpoints of an equilateral
triangle as its vertices. The initial point is the midpoint of the bottom edge. The curve that
joins every second vertex of the starting curve is the triangle in the middle. We view the starting
curve as passing through two edges of each of the three outer triangles.

We only have to describe for one of these triangles how the next iteration is obtained. We will
obtain curves that always run through two edges of each triangle, and the basic iteration can
always be applied. If we join every second vertex of the resulting curves then we obtain the
injective approximations of the Sierpinski Curve. 

The basic iteration step, for one triangle: \hfill\break
First add the two midpoints of the traversed edges of the triangle. Two more points are added,
one over the first and one over the last of the four subsegments. The points  lie in the inside of
the traversed triangle and they are the tips of isocele triangles whose base is the first, resp. the
last, of the four subsegments. In the case of the Sierpinski Curve these isocele triangles are
in fact equilateral. If the parameter $bb$ is smaller than $0.5$ then the height of the isocele
triangle is by the factor $bb/0.5$ smaller than the height of the equilateral triangle -- thus avoiding
the creation of double points of the approximation. \hfill\break
The iterated polygonal curve joins the initial point of the first edge to the first tip, continues to
the first edge-midpoint, passes through the vertex of the original triangle to the second
edge-midpoint, continues through the second tip and ends at the final point of the last segment.
The iterated polygonal curve traverses three triangles, two edges in each. Therefore the iteration 
step can be repeated.

The default morph in 3DXM varies $bb$ from $1/3$ to $1/2$ thus joining the first triangle contour
by a family of continuous (and injective) curves to the Sierpinski Curve.

H.K.
%\goodbreak\eject
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